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Higher rank instantons sheaves on Fano threefolds

Gaia Comaschi, Daniele Faenzi

TL;DR

The paper extends the theory of instanton bundles from rank 2 to higher rank on smooth Fano threefolds with Picard rank one, showing that the topological data of an (n,k)-instanton is governed by the pair (n,k) with k bounded below by a genus- and index-dependent k_0^n. It develops inductive constructions that produce μ-stable, unobstructed higher-rank instantons for all k ≥ k_0^n, starting from rank-2 ’t Hooft bundles and using extensions by minimal instantons or by F_0, depending on i_X. For even index, the base cases include P^3 and Del Pezzo threefolds, while for odd index, quadrics and index-1 primes are treated with genus-dependent minimal charges and deformation arguments; the restriction to K3 surfaces and Lagrangian geometry in moduli spaces are established. The work also exploits Kaditz Kuznetsov components to relate instantons to curves via acyclic extensions and monads, yielding monadic presentations and canonical resolutions that unify higher-rank instantons across several Fano families. Overall, it provides comprehensive existence, stability, and structural results for higher-rank instantons, along with categorical and geometric tools that connect instanton moduli to derived categories and curve-based data.

Abstract

We define instanton sheaves of higher rank on smooth Fano threefolds X of Picard rank one and show that their topological classification depends on two integers, namely the rank n (or the half of it, if the Fano index of X is odd) and the charge k. We elucidate the value of the minimal charge k0 of slope-stable n-instanton bundles (except for Fano threefolds of index 1 and genus 3 or 4), as an integer depending only on the genus of X and on n and we prove the existence of slope-stable n-instanton bundles of charge k greater than k0. Next, we study the acyclic extension of instantons on Fano threefolds with curvilinear Kuznetsov component and give a monadic description when the intermediate Jacobian is trivial. Finally, we provide several features of a general element in the main component of the moduli space of intantons, such as and generic splitting over rational curves contained in X and stable restriction to a K3 section S of X, and give applications to Lagrangian subvarieties of moduli spaces of sheaves on S.

Higher rank instantons sheaves on Fano threefolds

TL;DR

The paper extends the theory of instanton bundles from rank 2 to higher rank on smooth Fano threefolds with Picard rank one, showing that the topological data of an (n,k)-instanton is governed by the pair (n,k) with k bounded below by a genus- and index-dependent k_0^n. It develops inductive constructions that produce μ-stable, unobstructed higher-rank instantons for all k ≥ k_0^n, starting from rank-2 ’t Hooft bundles and using extensions by minimal instantons or by F_0, depending on i_X. For even index, the base cases include P^3 and Del Pezzo threefolds, while for odd index, quadrics and index-1 primes are treated with genus-dependent minimal charges and deformation arguments; the restriction to K3 surfaces and Lagrangian geometry in moduli spaces are established. The work also exploits Kaditz Kuznetsov components to relate instantons to curves via acyclic extensions and monads, yielding monadic presentations and canonical resolutions that unify higher-rank instantons across several Fano families. Overall, it provides comprehensive existence, stability, and structural results for higher-rank instantons, along with categorical and geometric tools that connect instanton moduli to derived categories and curve-based data.

Abstract

We define instanton sheaves of higher rank on smooth Fano threefolds X of Picard rank one and show that their topological classification depends on two integers, namely the rank n (or the half of it, if the Fano index of X is odd) and the charge k. We elucidate the value of the minimal charge k0 of slope-stable n-instanton bundles (except for Fano threefolds of index 1 and genus 3 or 4), as an integer depending only on the genus of X and on n and we prove the existence of slope-stable n-instanton bundles of charge k greater than k0. Next, we study the acyclic extension of instantons on Fano threefolds with curvilinear Kuznetsov component and give a monadic description when the intermediate Jacobian is trivial. Finally, we provide several features of a general element in the main component of the moduli space of intantons, such as and generic splitting over rational curves contained in X and stable restriction to a K3 section S of X, and give applications to Lagrangian subvarieties of moduli spaces of sheaves on S.

Paper Structure

This paper contains 22 sections, 37 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Preliminaries
  4. Rank 2 instantons
  5. Instanton bundles of higher rank
  6. Instantons on Fano threefolds of even index
  7. Instantons of high rank on ${\mathbb P}^3$
  8. Instanton bundles of high rank on Del Pezzo threefolds
  9. Instantons on Fano threefolds of odd index
  10. Instantons of high rank on three-dimensional quadrics
  11. Instantons on Fano threefold of index 1
  12. Rank 2 instanton bundles
  13. Instanton bundles of higher rank
  14. Stability of restriction to K3 surfaces
  15. Fano threefolds with curvilinear Kutznetsov component
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $E$ be an instanton sheaf. Then the following hold:

Theorems & Definitions (73)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 63 more